Question: Factor the following expression: $-5$ $x^2+$ $18$ $x+$ $35$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-5)}{(35)} &=& -175 \\ {a} + {b} &=& & & {18} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-175$ and add them together. Remember, since $-175$ is negative, one of the factors must be negative. The factors that add up to ${18}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${25}$ $ \begin{eqnarray} {ab} &=& ({-7})({25}) &=& -175 \\ {a} + {b} &=& {-7} + {25} &=& 18 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-5}x^2 {-7}x +{25}x +{35} $ Group the terms so that there is a common factor in each group: $ ({-5}x^2 {-7}x) + ({25}x +{35}) $ Factor out the common factors: $ x(-5x - 7) - 5(-5x - 7) $ Notice how $(-5x - 7)$ has become a common factor. Factor this out to find the answer. $(-5x - 7)(x - 5)$